Calculus visual reference

Common Derivatives Chart

A derivative rule tells you how a function's output changes as its input changes. Identify the outer structure first—sum, product, quotient, or composition—then apply the matching rule.

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Common derivatives chart showing constant power product quotient chain exponential logarithmic and trigonometric derivative rules
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How to read it

Use the chart as a decision tool.

A printable derivative rules chart with constant, power, product, quotient, chain, exponential, logarithmic, and trigonometric derivatives. Read the label first, identify the matching structure in your problem, then confirm that any stated conditions apply before substituting values.

01

Core rules

These rules combine to differentiate larger expressions.

Constant

ddx[c]=0\displaystyle \frac d{dx}[c]=0

Power

ddx[xn]=nxn−1\displaystyle \frac d{dx}[x^n]=nx^{n-1}

Product

(fg)′=f′g+fg′\displaystyle (fg)'=f'g+fg'

Quotient

(fg)′=f′g−fg′g2\displaystyle \left(\frac fg\right)'=\frac{f'g-fg'}{g^2}

Chain

ddxf(g(x))=f′(g(x))g′(x)\displaystyle \frac d{dx}f(g(x))=f'(g(x))g'(x)
02

Common functions

Assume each expression is in its differentiable domain.

Exponential

ddxex=ex\displaystyle \frac d{dx}e^x=e^x

Logarithm

ddxln⁡x=1x\displaystyle \frac d{dx}\ln x=\frac1x

Sine / cosine

(sin⁡x)′=cos⁡x,(cos⁡x)′=−sin⁡x\displaystyle (\sin x)'=\cos x,\quad(\cos x)'=-\sin x

Tangent

(tan⁡x)′=sec⁡2x\displaystyle (\tan x)'=\sec^2x
How to remember it

Name the outer operation first

  1. 1Sums differentiate term by term.
  2. 2Products and quotients need their own rules; do not differentiate each factor separately.
  3. 3A composition signals the chain rule: outside derivative times inside derivative.
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